Predicting Call Option Prices Using Regression

Models

Munira Shahir

July 24, 2014

Abstract

One method to consists of predict call option prices is the Black-Scholesequation. However, that is utilizing for predicting European call optionprices not American call option prices. The work done consisted of util-itizing regression models to predict call option prices utilizing known in-formation from the New York Stock Exchange (NYSE)

1 Introduction

In finance, an option is a financial contract in which the buyer has the right,but not the obligation to buy or sell an agreed upon price on or before a spe-cific data. The contract to buy an option is a call option. The contract tosell an option is a put option. These two are related by put-call parity. Theagreed upon price is the strike price. The date to decide whether or not theoption by is the exercise date. There are two types different styles of optionsAmerican and European. Owners of a European style options may exercisethem only at expiration. Owners of American style options may exercise themat and before expiration. For predicting the price of European options, BlackScholes equation is used by some investors. However, this is only useful as anapproximation for the option price. Also, there isn’t an equivalent equationto the Black-Scholes equation for predicting American option prices. This thanleads into developing regression models to predict call option price. This is doneutilizing linear regression models and non-parametric regression models. Thenon-parametric regression models consist of generalized additive model (GAM),projection pursuit regression (PPR), regression trees, and random forest. Thepredictors consist of the strike price of the option, the current price of the stock,expiration time and the historical volatility, which is how much the option pricevaried over the last two months. The response is the option price. The data istaken from the New York Stock Exchange (NYSE).

2 Methods

2.1 Linear Regression Models

Linear regression models consist of modeling the response as a linear functionof the predictors. y = β0 +

∑βixi + ε. Where y is the predicted response,

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the βi’s are the parameters, the xi’s are the predictors, and ε is the residualerror. One of the ways to determine if a linear regression model would work isby examining a scatterplots of the predictors and the response,in addition totheir log transformations.

2.2 GAM Models

GAM models consist of modeling the response as a linear combination of func-tions of a single predictor. y = β0+

∑βifi(xi)+ε This model tested all possible

combination of predictors for determining the response, in addition to their logtransformations.

2.3 PPR Models

PPR models are a generalization of GAM models. In this models functions of alinear combination of predictors utilized to predict the response, in addition totheir log transformations. y = β0+

∑βifi(α

Ti x)+ε Where y is the response,βi’s

are the parameters, αi’s are the projection direction vectors, and ε is the residualerror.

2.4 Regression Tree Models

Regression Trees consist of partioning the data into intelligently chosen subsets,and in each subset the response will be modeled as the mean. This model testedall possible combination of predictors for determining the response, in additionto their log transformations.

2.5 Random Forest Models

The Random Forest model consist of generating 500 new data sets from theoriginal data set. Then a tree is constructed on each of these data sets. Thefinal prediction is then obtained by averaging predictions from all 500 trees. Thiswas tested for up to two predictors, in addition to their log transformations.

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3 Plots

3.1 Scatterplots

3.1.1 Current Price vs Option Price

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3.1.2 Current Price vs log(Option Price)

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3.1.3 log(Current Price) vs Option Price

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3.1.4 log(Current Price) vs log(Option Price)

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3.1.5 Strike Price vs Option Price

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3.1.6 Strike Price vs log(Option Price)

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3.1.7 log(Strike Price) vs Option Price

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3.1.8 log(Strike Price) vs log(Option Price)

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3.1.9 Historical Volatility vs Option Price

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3.1.10 Historical Volatility vs log(Option Price)

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3.1.11 log(Historical Volatility) vs Option Price

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3.1.12 log(Historical Volatility) vs log(Option Price)

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3.1.13 Expiration Time vs Option Price

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3.1.14 Expiration Time vs log(Option Price)

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3.1.15 log(Expiration Time) vs Option Price

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3.1.16 log(Expiration Time) vs log(Option Price)

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3.2 PPR Models

3.2.1 log(option price) current price + strike price

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3.2.2 log(option price) log(current price)+log(strike price)

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3.3 Random Forest Models

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4 Results

4.1 Linear Regression Models

It’s apparent from the scatterplots that there isn’t a linear relationship betweenthe expiration time and the option price. A lack of a linear relation ship betweenthe historical volatility and the option price. With regard to the current strikeand strike price, they both have a slight linear relationship with the optionprice. However, it’s apparent that both of them don’t have a strong enoughlinear relationship with the option price. As a result, linear regression modelsaren’t a good choice for modeling option prices.

4.2 GAM Models

A number of the tested models failed at predicting the response. There weresome models that initially appeared to work, however, there when tested onanother data set the model failed at predicting the response. This was do tothe model being heaviliy overfitted.

4.3 Projection Pursuit Regression Models

The ”good” models worked best for predicting smaller option prices, This isdue to the data posssesing more information for the smaller option prices. Thetwo models both consist of first predicting the log of the option prices. Thepredictors used in the first model are the current price and the strike price. Thepredictors used in the second model consists of the log transformations of boththe current price and the strike price of the stock in question.

4.4 Regression Tree Models

A number of the tested models failed at predicting the response. There weresome models that initially appeared to work, however, there when tested onanother data set the model failed at predicting the response. This was do tothe model being heaviliy overfitted.

4.5 Random Forest Models

The model that worked tended to underestimate the option price. This modelconsisted of predicting the log transformation of the option price based on thelog transformations of both the current price and strike price of the stock inquestion

5 Conclusions

In conclusion, the best models for predict call option prices are PPR and Ran-dom Forest. Future work would consist of improving the PPR models, usingdifferent non-parametric models, and factoring the volume of options sold fromthe amount avalable.

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